First we prove injectivity. Let \(\sigma \)  and \(\tau \)  be
\(Z\) -separating term orderings for \(I\) , write \(G=\lbrace g_1,\dots ,g_r\rbrace \) 
for the reduced \(\sigma \) -Gröbner basis of \(I\) , and write
\(H=\lbrace h_1,\dots ,h_{r^{\prime }}\rbrace \)  for the reduced \(\tau \) -Gröbner basis of \(I\) .
Here we may assume that \(r\ge s\)  and \(r^{\prime }\ge s\)  and
\(\mathop {\rm LT}\nolimits _\sigma (g_i) = \mathop {\rm LT}\nolimits _\tau (h_i) = z_i\)  for \(i=1,\dots ,s\) .
Suppose that the map \(\Gamma _Z\)  satisfies
\(\Gamma _Z(\overline{G}) &\;=\; \lbrace \, (\mathop {\rm LT}\nolimits _\sigma (g_{s+1}), g_{s+1}),\; \dots ,\;(\mathop {\rm LT}\nolimits _\sigma (g_r),g_r) \,\rbrace \\&\;=\; \lbrace \, \mathop {\rm LT}\nolimits _\tau (h_{s+1}), h_{s+1}),\; \dots ,\;(\mathop {\rm LT}\nolimits _\tau (h_{r^{\prime }}),h_{r^{\prime }}) \,\rbrace  \;=\; \Gamma _Z(\overline{H})\) 

Then we see that \(r=r^{\prime }\)  and \(\mathop {\rm LT}\nolimits _\sigma (g_i) = \mathop {\rm LT}\nolimits _\tau (h_i)\) 
for \(i=s+1,\dots ,r\) , after possibly renumbering \(h_{s+1},\dots ,h_r\) 
suitably. By the lemma, we get \(G=H\) , and hence \(\overline{G}=\overline{H}\) .
Next we prove surjectivity. Let \(\tau \)  be a term ordering on \(K[Y]\) ,
and let \(H = \lbrace h_1,\dots ,h_t\rbrace \)  be the reduced \(\tau \) -Gröbner basis of \(I\cap K[Y]\) .
Using the theory of Gröbner fans (cf. [1]}), we may assume that
\(\tau = {\rm ord}(W)\)  with a matrix \(W\in \mathop {\rm Mat}\nolimits _{n-s}(\mathbb {Z})\) .
By assumption, the set \(\mathop {\rm GFan}\nolimits _Z(I)\)  is not empty.
So, let \(F=(f_1,\dots ,f_s)\)  be a coherently \(Z\) -separating tuple of polynomials in \(I\) .
Now we construct a matrix \(V\in \mathop {\rm Mat}\nolimits _n(\mathbb {Z})\)  which defines a
term ordering \(\sigma = {\rm ord}(V)\)  in \(\mathbb {T}^n\)  as follows.
First we put the matrix of size \(s\times s\)  defining the ordering
\({\tt DegRevLex}\)  on \(K[Z]\)  into the columns of \(V\)  corresponding to \(Z\) 
(see [2]}, Definition 1.4.7 and Proposition 1.4.12)
and zeroes into the other columns. This fills the first \(s\)  rows of \(V\) .
In the lower \(n-s\)  rows, we put \(W\)  into the columns of \(V\)  corresponding to \(Y\) 
and zeroes into the columns corresponding to \(Z\) .
The resulting term ordering \(\sigma \)  satisfies \(\mathop {\rm LT}\nolimits _\sigma (f_i)= z_i\) 
for \(i=1,\dots ,s\)  and \(\mathop {\rm LT}\nolimits _\sigma (h_j) = \mathop {\rm LT}\nolimits _\tau (h_j)\)  for \(j=1,\dots ,t\) .
Therefore the reduced \(\sigma \) -Gröbner basis of \(I\)  is of the form
\(G=\lbrace  z_1 - g_1,\dots ,z_s - g_s,\, h_1,\dots ,h_t\rbrace \)  with \(g_1,\dots ,g_s \in K[Y]\) 
and we get \(\Gamma _Z(\overline{G}) = \overline{H}\) , as desired.
